One of the most curious arguments for the existence of God has been presented by St. Anselm, René Descartes, and many other theologians throughout the centuries: the Ontological Argument. The classical formulation of the argument is (1):

1. God is that entity than which nothing greater can be conceived.2. It is greater to be necessary than not.3. God must be necessary.4. God necessarily exists.

Perhaps the most challenging formulation of the argument is presented today by Alvin Plantinga. Dr. William Lane Craig presents Plantinga's argument as (2):

1. It is possible that a maximally great being exists. 2. If it is possible that a maximally great being exists, then a maximally great being exists in some possible world. 3. If a maximally great being exists in some possible world, then He exists in every possible world. ($$) 4. If He exists in every possible world, He must exist in the actual world.

I will discuss this particular formulation at length in this article.

The Classical Take

A brief statement about the classical version of this argument is necessary, particularly about the necessity of "necessary" being an inherently positive quality in and of itself, without regards to its referent in reality. This is not entirely clear; a fantastic counterexample would be certain events in the context of human history, which as an A-time theorist I hold to be necessary facts of existence. Suppose, for instance, that Adam and Eve existed and chose to Fall. Then, unless one is a high-Calvinist, the necessity (by asssumption) of the Fall would be a negative quality, as opposed to a positive one, as the action in the Fall brought death and damnation to Adam, Eve, and subsequently, to all of us. Therefore, it cannot be established that necessity qua necessity is an inherently positive quality of existence.

Other refutations exist of this presentation, from skeptics to believers. Hume famously rejected the argument by stating that it was logically possible to conceive the nonexistence of every entity in reality, i.e. it is logically possible to conceive that the truth value of the existence of every particular entity in the actual world is equal to "false." Geisler and Corduan endorse this objection (3).

Plantinga's Take

Plantinga opens the playing field to the set of all possible worlds. In his presentation, we are asked to imagine every logically possible world, where a possible world is defined to be a world (or state of existence) such that the set of all logical facts describing the world

p1^p2^p3^...^pn

exists without internal contradiction. Now, personally, I agree with the rather minority viewpoint that the only possible world is the actual world, given that I accept the necessity of entities in reality as my primary philosophical axiom (4). But for the sake of argument, I will accept a multiplicity of possible worlds, assumed in this case to be infinite (5). Although counterintuitive if we view Craig's presentation (6) of possible worlds as consisting only of a finite set of facts, we are assuming that the set of facts in one possible world can differ from the set of facts in another possible world arbitrarily. Let's begin the critique with these understandings.

Craig asserts all premises save premise #1 is "relatively uncontroversial," a point which I disagree heavily and will touch upon later. He then goes on to establish a priori warrant for premise #1, stating that the intuitive concept of an omnipotent, omnibenevolent, and omniscient being must be logically incoherent to invalidate the premise. Bill first attempts to show how typical objections along the "maximally great island" lines fail, asserting "there could always be more palm trees and native dancing girls." While true, I sort of have an ill feeling that Craig's criticism is misplaced: certainly an island completely filled with dancing women to the point where nobody could move anywhere but into the ocean would be less great than if four or five women were there to greet me with some freshly cut coconuts.

Bill states the stronger criticism that such concepts are relative to the observer; perhaps, as Bill says, another person would prefer a full resort while another an empty desert island. Indeed, the person in question could be a woman who would prefer men on the island, or, in my case, some other tropical fruit apart from coconuts given my distaste for them. But that doesn't disprove the notion that a maximally great tropical island is logically possible for all humanity (who are interested in these things to begin with). Perhaps the island could contain several resorts sorted for particular tastes, and could contain areas of desertion where those who prefer to be away from civilization could relax maximally. It may be so that not every desire is satisfied by all of our prospective visitors, but the fact remains that a resort island built to maximize every interested person's preferences and leave everyone at least happy that they came is not necessarily impossible logically due to relativity in preference.

A discussion follows regarding quasi-maximal beings. Let's suppose that a logically possible world W1 exists where the maximal being exists and where Fred Sanford, say, is born with omnipotence and omniscience, but not omnibenevolence (Bill offers an example of a quasi-maximal being lacking knowledge of future events). This being may be derived in the same method as the maximal being: given the establishment of a maximal being, we may use the same logic to establish a quasi-maximal being, i.e. the necessary existence in a possible world of a being "one step down" from our maximal one.

Craig correctly states that the maximal being could choose not to create the quasi-maximal being with His creation powers, but why do we presuppose that the maximal being created? Given these premises, it is not logically impossible that a quasi-maximal being exists; perhaps in a possible world Fred Sanford and God existed side-by-side, with the only difference between the two (apart from conscious separation) being that Fred had a bit more heartburn. In this presentation, our quasi-maximal being would be uncreated, as our maximal being is. As another side note, it is possible, logically, that a quasi-maximal being created the maximal one; there is no premise that states that to be created is less in maximality than to have existed in all states of our logically possible world, only that the maximal being's existence in this world is necessary. It is my charge, then, that the challenges stand and that the a priori concept of a maximal being still presents the incoherency that Craig assumes to have been refuted. (**)

Craig then goes on to discuss an a posteriori establishment of Premise #1, but Craig treads carefully here: "I remain uncertain of this argument ... which would require us to reject various nominalistic alternatives to conceptualism such as fictionalism, constructabilism, figuralism, and so forth. Still, prominent philosophers such as Plantinga have endorsed it." (7) This concern is brought forth from Plantinga's a posteriori establishment via means of grounding abstractions metaphysically n the mind of some being, since, as Plantinga argues, they cannot be established in our own.

It took me a while to understand Craig's persistent stomachache over this (ultimately leading to the aforementioned confession of Craig's doubt about the Ontological Argument), but then I remembered that this sort of argumentation from the supposed inability to ground concepts in reality is part of Bahnsen's Transcendental Argument for the Existence of God (TAG).(8)Bahnsen, and Plantinga himself, are generally Reformed, and come from a very rationalist and highly skeptical (in the philosophical sense) worldview that Craig, an evidentialist and classical apologist, tends to shun (TAG, for instance, appears nowhere in Craig's book). I side with Craig here, but a post about this will have to wait.

There is one more difficult premise, however, that Craig accepts without discussion, but that Plantinga elsewhere (9) both understands and attempts to correct: the inductive premise #3, i.e. the premise that if this maximally great being exists in some logically possible world, He exists in all possible worlds, including our actual one.

Loosely, Plantinga describes "maximally excellent" as necessarily including the three omni-'s: benevolence, potence, knowledge. If this Being enjoys maximal excellence in all possible worlds, then this Being is "maximally great." Plantinga wishes to establish premise #3 by establishing that a maximally greatBeing exists in at least one possible world.

As a side quip, why are the three omni's properties of a maximal being and considered to be tops in evaluating excellence? If I were omniscient - and I'm sorry to go to toilet humor, but you see what I mean - I'd see everyone poop. Not excellent! All kidding aside, who would want to know every detail of the Holocaust, particularly if one were additionally omnibenevolent and omnipotent (but doing the Arminian thing of letting people act on their own power of choice)? But I'm being mean and too speculative, so I will grant Plantinga that the three qualities give a being a great degree of excellence exceeding beings which do not possess these qualities. I'll also be nice and grant that only one being possessing these three qualities can exist in logical possibility in a given possible world.

How are we to establish values for these omni-qualities and for the rest of the qualities of our theoretical being, or any being, for that matter? Plantinga proposes a number (assumed to be bigger than or equal to zero) describing the "excellence" of an extant entity in some possible world, and asks us to sum this excellence number over all possible worlds where the entity in question exists. Such a sum is taken only over the possible worlds containing the assumption of x's existence, for, the concept presupposes existence and therefore an "excellence number" has no value in a world where the truth value of x's existence is false. It's not even zero - it would be, if you remember Algebra, like taking the square root of a negative number while working only with reals. It's simply outside the domain!

But still, how can one metaphysically quantify "excellence" for any particular entity? Nowhere do either Plantinga or Mears propose such a means, but I'll propose a concrete definition: an excellence number represents the number of entities which the entity "x" is, in a sense, "better in more individual respects" than. This would allow God, as omniscient, omnipotent, and omnibenevolent, to have an excellence number equal to the (finite) sum of entities in reality, playing on the undiscussed intuitive notion here that a being with the three omni's is better than all the rest of the entities in reality.

So, taking the sum of excellence over all possible worlds well-defines a function F(x) as such:

F(x) = W1 + W2 + W3 + ... + Wn

This is the "greatness function," as greatness, remember, is to be taken as a representation of the excellence over ALL WORLDS where x exists. Here n represents the number of worlds where x possibly exists. Taking the limit as n goes to infinity gets our "greatness number" for an entity possibly existing in an infinite number of worlds. We assume our case for a maximally great being must exist in an infinite number of possible worlds, then, since if He were to exist in only a finite number of worlds the chance of His existence in the actual world, although logically possible, would be (letting k be the total number of possible worlds)

lim k->infinity (n/k) = 0,

a point which Plantinga and his critic Mears miss in their respective papers.

The conclusion of Plantinga's case for premise #3 is that

lim F(x), x:= "God"

is a number greater than all other greatness valuations F(x') taken over any arbitrary non-Godly entity x' that exists in any possible world.

Here's one killer, built on the idea formulated by Mr. Mears. Note here that F(x) (God's greatness) must be finite to make any sense. For, if

lim F(x) -> infinity,

then the coherency of the maximally great being vanishes logically (10). Therefore, this greatness must be a concrete number. Plantinga's reasoning for this has been refuted above, but Plantinga is still correct for the reason I give in the footnote: natural numbers themselves do not have a greatest upper bound, and although they cannot be used to describe actual metaphysical quantity as e.g. increasing girls indefinitely on our island, the possibility of the comparison with our "greatness function" exists inherently in the definition. Therefore, if it can be demonstrated that the function evaluates to infinity, then God's greatness, e.g. his summation of his "excellency rating" over every world in which He exists, becomes logically incoherent, collapsing the argument. (&&)

Since the existence of God over a finite subset of the (infinite) set of logically possible worlds leads to a zero-probability of His actual existence, as mentioned earlier, we must take the infinite sum. Following my coherent definition of what it means to have excellence, i.e. an integer representing the number of entities that "x" is "more excellent than" in the intuitive sense, then only one logically possible world has an excellence rating of zero for God - the one where only He exists. Therefore there are an infinite number of positive integer representations of excellence-ratings in possible worlds. Letting W1 = 0 (our excellency rating in the one world where only God exists), we have, then:

Even if Plantinga would object to my definition, he must at the very least represent a function whose infinity-limit tends toward zero if we wish for the summation function F(x) to yield a finite number (to wit, the sum of 1/n^2, n from 1 to infinity, is a finite positive number, and 1/n^2 itself tends to 0 as n rises without bound, fitting our bill).

But this implies that God's excellence by any definition Plantinga wants to use, when ordered, must tend to zero and therefore gets arbitrarily small - and furthermore, must do so in a matter that still yields a finite sum (for, sum 1/n, n from 1 to infinity, is still infinity despite the fact that limit 1/n ->0 as n tends infinitely). As this is intuitively counter to our notion that these values of excellency ought to be large, it seems a tough challenge indeed to define an excellency valuation which both tends to zero when ordered, is such that the sum is still a finite number, and is so that, despite being damned near zero for all but a finite number of possible worlds, it is still greater than all other entities' excellency evaluations in each of those particular possible worlds. And that's if a coherent definition of how to evaluate an "excellency rating," apart from the one I offered, is first given!

One final point, unobserved by neither Plantinga nor Meirs - we have, in the latter part of this paper, only discussed a sum over an infinite number of possible worlds. This is the direct case allowed by Plantinga's presentation. All it would show even if every premise is true and justified is that God exists in an infinite number of possible worlds, and not the actual one, and at most with his definitions this presentation can only establish a probability of God's existence, rather than a logical certainty. (##)

I conclude that Plantinga's presentation of the Ontological Argument has been refuted, pending critiques, comments and discussions from the readers of this blog. I am looking forward to an engaging discussion.

And am hoping nobody fell asleep because of the math. :)

-Darrin

Sources and Notes

(1) St. Anselm. Proslogion, Ch. 2. Retrieved from Wikipedia. See also criticisms of this presentation in Dr. Corduan's response to this post.

Corduan notes in the commentary to this post: "Of course, I'm not sure we mean it in the same sense as Hume did, but it's always nice to see it when people realize that there is an unspoken assumption underlying the ontological argument, namely that something exists." (Retrieved 1/19/09, with thanks to Dr. Corduan).

(4) See e.g. Rand, Ayn, "The Metaphysical Versus the Man Made."

(5) Mears, Tyrel. "Sympathy for the Fool," p. 87. If one buys the premise of "possible worlds" in the first place, it logically follows that the number of "possible worlds" is infinite. For, it could be logically possible for a world outside of a proposed finite set of worlds exists where I swivel my chair completely around here at my desk; one where I swivel halfway around; one where I swivel one-forth around; etc. leading immediately to an infinite set of possible worlds if one does not, as I do, hold the necessity of entities and action in the actual world as a foundational premise.

(6) Craig 183.

(7) This quote, as well as the preceding analysis and a bit more proceeding this footnote, are discussed in full on (Craig) pp. 184-189.

(8) Bahnsen mentions the inability of the non-Christian to ground concepts in "The Great Debate" versus Gordon Stein, a debate which I am planning to review in the near future, but Bahnsen's presentation of TAG leans more heavily on the inability to ground the process of reasoning. Personally, I believe the two are interrelated, if not equal processes, and at the very least the former precedes the latter inclusively (see e.g. Ayn Rand, "Introduction to Objectivist Epistemology," or earlier Wittgenstein). See also Gordon Clark's Youtube lectures regarding his critique of Empiricism; both Clark and Bahnsen are free for anyone to watch on that website.

(9) The rest of Plantinga's quotes have been retrieved from Mears' paper, documented earlier (once again here:http://aporia.byu.edu/pdfs/mears-sympathy_for_the_fool.pdf). All quotes from Plantinga are properly annotated in the work, and I have found no indication that Mr. Mears misuses or presents in a nonobjective manner any material written by Dr. Plantinga. The references to Mr. Mears himself which follow from this annotation until the end of my paper are taken, for the most part, from pp. 83-91.

(10) Plantinga, Alvin. God, Freedom, and Evil. P. 91. Plantinga uses the objection mentioned by Craig, about the paradise island that can indefinitely gain more girls and coconuts and get better. But this has been refuted earlier; Plantinga's necessary inherent maximum is present for this concept, as well. However, for the (entirely epistemological) natural numbers, e.g. 1, 2, 3, 4, 5, ..., there is in fact no "greatest number," since infinity is not recognized as a member of the natural number set (11). Thus, with respect to the natural number set, the idea of a "maximal natural number" is incoherent, which properly captures the idea intuited by Craig and Plantinga in the "add more girls!" objection, while not escaping the context of practicality.

(11) See e.g. Royden's Real Analysis, Ch. 1.

(12) ibid.

($$) There is an issue with Plantinga's modal logic presentation of this premise that goes undiscussed here, due to my unfamiliarity with the subject (it's been over a decade since I've seen it!). Refer to http://barefootbum.blogspot.com/2008/05/modal-logic.html for a criticism on this subject. (Thanks to The Barefoot Bum and Dr. Corduan for bringing this issue up in the commentary of this post.)

(**) As a side note, it may be stated that if it can be shown that two omnipotent beings exist in a logically possible world, where one of the two beings is not omniscient and one is, and if it can be logically possible that the nonbenevolent Being would wish to engage the benevolent Being, the dual omnipotence assumed in this case would render a logical impossibility. Perhaps via traveling down the quasi-omniscient chain we may always find such a "Fred Sanford" through inductive establishment on the basis of the establishment of the existence of the maximal being. This would demonstrate the necessary logical incoherency of omnipotence, and would be a logical disproof of God in any possible world.

This is merely postulation, however, since none of my premises have been established in this paragraph; but should they turn out logically sound, then assuming the existence of a maximal being in all logically possible worlds may logically lead to the existence of a quasi-maximal omnipotent, omniscient, but anti-benevolent being in every possible world, who, through the definition of benevolence, would both desire to defeat one another as their highest priority. But since both are omnipotent and omniscient, they both can and can not defeat one another; we would then establish the non-existence of God through the impossibility of the contrary.

(&&) - If we accept this quantity to be possibly infinite despite the intuited objection from the natural number comparison, we run into problems: if any other being had a "greatness valuation" F(x') = infinity, it would be impossible to quantify x' and x in greatness relation (the infinite is countable in any case by definition). As we shall see later in the paper, all it will take is for some other being x' to exist with quality in an infinite number of possible worlds to get an infinite greatness evaluation - let's say that the Devil, for instance, exists in an infinite subset of possible world, and that his valuation of excellency, given the Devil's immense powers described Biblically, is always greater than zero in each world. By the argument which follows this annotation, the Devil would, in this case, have a greatness valuation of infinity, equal to God's; even if the Devil existed in "less" worlds than God (but still existed in an infinite subset of possible worlds), we'd have to equate the Devil and God with greatness, and I don't think we want to say that.

(##) - Again, Plantinga's summation is defined over an infinite number of possible worlds, not all possible worlds. Plantinga has made the mistake in assuming that since the number of possible worlds is infinite, then the sum taken over infinity covers it. But this may not be so. For, by mathematical postulate, we may well-order all the possible worlds in this set; it might be the case that God exists in all odd-numbered worlds W1, W3, W5, W7, W9, .... so that even if Plantinga destroys my case but fails to establish how God's existence in an infinite number of worlds entails His existence in all worlds, we might have that

W1+W3+W5+W7+W9+ ... = F(x)

is a finite number indeed, establishing coherency and validating the argument, but still leaving the evens out of the consideration (the ones where it is possible God does not exist). This leaves only, in this assumed case, the probability of God to be N/2N = 1/2 for our actual world (assuming we don't know which possible world-number it is) even if we assume all of Plantinga's case as otherwise true and valid.

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comments:

I have to confess that I got completely lost in the formal logic and math as my mind isn't all that analytical. Would this rewording of the argument show the fallacy?:

1. It is possible that an all-encompassing red sky exists.2. If it is possible that an all-encompassing red sky exists, then an all-encompassing red sky exists in some possible world.3. If an all-encompassing red sky exists in some possible world, then it exists in every possible world (being all-encompassing, you see).4. If it exists in every possible world, it must exist in the actual world.

and, 5. If you look up and see a blue sky then you're not seeing things correctly. It's red, dammit!and, 6 (because this is Bill Craig). Give your heart to the Red Sky and you will experience an inner transformation that will change your life. If I have failed to convince you it is my weakness, not the weakness of the Red Sky.

Using Plantinga's formulation John Hick argued that a maximally evil being exists in every possible world too. Since this same type of reasoning leads to opposite conclusions the reasoning is false without having to bother as you did, Darrin.

Talking about maxima and minima and omni- qualities are well suited to mathematics but I have a hard time believing that they have any relevance to real-world beings.

What does it mean for any being to be omnibenevolent or omniscient? How could we tell if we thought we'd found an omni being?

Why does Craig/Plantinga think that a maximally anything being exists? How could it evolve? How could even a stepwise process of refinement (which evolution is not!) ever achieve maximality? We may be able to conceive of "omniscience" in abstract but what reason do we have to think it's possible for any being to ever achieve this?

It sounds like this argument assumes that God is omni-everything, that God can exist and that God must exist and uses this to conclude that God does exist for surely the only way to side-step around these vital issues is to just say that this omni entity is "necessary" and thus dodge any responsibility for demonstrating how it arose.

Yet what is "necessary"? I've only taken a couple years of physics in University and read some pop physics science but I've never seen any physics theory using any necessary being. Why is it that only armchair theologists talk about what's necessary for the universe in puffy, ill-defined terms and the people that actually understand our physical cosmological theories do not? Could this "necessary" being not be necessary after all?

In my philosophy courses we would frequently prove theories to be wrong by starting with a set of assumptions and then deriving a contradiction. I see a lot of potential contradictions here. Perhaps the only thing that the ontological argument demonstrates is that the premises are false and that it is not possible that a maximally great being exists.

Another thing I see in the sciences is a nearly obsessive effort to disprove one's own theories. When Penzias and Wilson first encountered radio noise they could not explain they did not publish and claim a great discovery though they must have thought about it. Instead they did everything they could to account for all other explanations they could think of, even the wildly improbable. In one case they thought it might be caused by pigeon droppings even though this was a long shot but they put in the time and effort to clean the droppings and eventually shot all the pigeons. They published only when they could stand up in front of a crowd of hostile, informed colleagues and present, knowing they had thought of everything that might undermine their theory.

I don't see even a glimmer of this sort of intellectual honesty from Plantinga or Craig. Everything is abstract as if it were referring to some mathematical domain yet this is never specified and they actually act like this represents the real world yet they haven't provided the first step towards verifying their ideas. My second year math professor would give this a fail (and my philosophy prof wouldn't touch it) so why does this pass with high honours in theology?

Dear Darrin and readers,Thanks for the reference to Norm Geisler's and my principle of the contingency of all finite entities in the universe. Of course, I'm not sure we mean it in the same sense as Hume did, but it's always nice to see it when people realize that there is an unspoken assumption underlying the ontological argument, namely that something exists. I'm afraid that the summary version of the ontological argument, as presented in the Wikipedia (your evident source for it), is neither historically nor conceptually very precise. The first descriptor, aliquid quo maius nihil cogitari potest does not use the word "entity" (ens), which Anselm deliberately avoided. I'm intentionally stating the Latin so that people can see how much we frequently import into Anselm's very carefully chosen phrases. That sentence is the opening premise of the supposed "first" argument by Anselm, whose decisive criterion is the distinction between the lesser greatness of existing in the mind alone and the greater greatness of existing both in the mind and reality. It is impossible to understand this distinction without a willingness to immerse yourself into an ancient form of metaphysics and see where this notion comes from before attempting a creditable criticism of it. There is nothing about "necessary existence" in the first argument. The notion of "necessary existence" emerges in the second argument, which is based on the inconceivability of the nonexistence of whatever is insurpassible in its being. This observation would apply even if we did not care to apply the term "God" to it, and, in fact, Anselm intentionally works up to why it should be called "God" subsequently. Here all we get is Sic ergo vere est aliquid quo maius cogitari non potest, ut nec cogitari possit non esse. Then, the next sentence heads a long section deriving the attributes of God from this "necessary being." In God and Other Minds, Plantinga picked one particular version of the cosmological argument, called it the locus classicus, shot a noisy blank at it, and declared it to be dead. I love the way you try to play the same dramatics with Plantinga's ontological argument. If you can find a hole in it, so much the worse for all other versions of the argument. That's pretty dicey isn't it? And doesn't it get even scarier if that's the approach you use to make the case for the non-demonstrability of God's existence simpliciter? I'm less sure than you are that you have demolished a particular argument by finding a hole in one version you have selected, let alone that doing so takes you on the way to demonstrating the non-existence of God.

I'm also disappointed to see that you are unaware as to the meaning of Plantinga's premise that "if this maximally great being exists in some logically possible world, He exists in all possible worlds, including our actual one." This has absolutely nothing to do with assigning numbers to greatness distributed over the available number of logically possible worlds. There is an issue here, but it has to do with the different versions of modal systems one can use in constructing or evaluating an argument. Planting is consciously accepting a system called "S5," which allows the premise to stand, but other systems, e.g. "T," "B," and "S4" would not allow that premise. What I'm getting at is this: As you are working through your project of "debunking Christianity," please make sure that you're not doing so at the cost of creditable scholarship. For many years, I and other teachers of Christian apologetics have tried to impress on our students that they need to study hard in order to make a solid case for the truth of Christianity, though I'm afraid we haven't always been successful. However, essays such as the one you posted appear to demonstrate that neither is it necessary. What use is solid Christian scholarship if non-Christians write essays such as the one you posted that make scholarship unnecessary? I have always said that if Christianity is true it should be able to withstand the hardest legitimate questions. Shouldn't the same criterion apply to a non-Christian world view? Thanks again for the reference. Have a good day.Win

I agree with most of the above analysis and I agree that Plantinga's argument fails for a multitude of reasons. Even he doesn't quite accept the conclusion of his own argument, as Beachbum points out in entry on the subject.

However, I tend to agree more with the classical objections with Geisler. There is no logical contradiction inherent in assenting to the proposition "God does not exist" as logic (in-and-of itself) makes no positive existential claims without entering existential premises. It is for this reason that we can enter the premise:

'It is logically possible that a maximally great God exists' or 'It is logically possible that a maximally great God does not exist',

yielding contradictory conclusions from premises that both seem relatively harmless at first glance and both of which seem to be able to be justified in ways similar to Craig's method.

In order for Plantinga's argument to be successful, he must instead talk about metaphysical possibility. He must be able to state as a metaphysical fact that it is possible that a maximally great God exists in some possible world. That is, there exists (modally speaking) a metaphysically possible world in which a maximally great God exists. But this is a positive existential claim which I would love to see demonstrated. Such a demonstration would require the entering of an existential premise and metaphysical principles. He must be able to conclude that there is no metaphysically possible world in which a maximally great God fails to exist, which is a mighty claim.

The metaphysical problem boils down to this:

1) If it is possible that a maximally great God exists, then a maximally great God exists.

1*) If it is possible that a maximally great God does not exist, then a maximally great God does not exist.

But this is virtually identical to the kinds of things that the Ontological Arguments of the past have claimed. Without a grounding in metaphysical fact, the argument makes the mistake of conflating conceivability with metaphysical possibility. For these reasons, I just don't see Plantinga's argument as much of an improvement over the other modal arguments given by Hartshorne and Malcolm.

"I have always said that if Christianity is true it should be able to withstand the hardest legitimate questions. Shouldn't the same criterion apply to a non-Christian world view?"

I'm not entirely sure why I do, but I do have some serious problems with this statement.

First, you are assuming that the claims of the existence of a Christian god are on equal footing as a positive claim with a default position that there is no such god or gods. Yes, your claim should be able to withstand the hardest legitimate questions, and if it cannot, then it is not a legitimate claim.

The end.

Of course, if we cannot say for sure whether or not there is a Christian god, just because no one has proved that there isn't one, that does not mean that there is one!

It appears to me that you are assuming that unless someone can prove that there is no Christian god, then your claims are credible. If this is not the case, please tell me what you meant when you said "shouldn't the same criterion apply to a non-Christian worldview?". Thanks.

That's not quite a rewording of the argument, according to Plantinga et. al. They would counter at point #3, more than likely, since it is logically possible on-the-face that a world exists with a blue sky.

Some logical workings in your presentation are akin to the "desert island refutation" objections that I wrote in regards to Craig and Plantinga.

Thanks for the commentary. I mentioned in the paper that Plantinga has clearly not defined a means by which one is to quantify excellence in a particular world. Perhaps he has in other works, but I have neither seen nor heard of any numerical gauge used. I proposed a gauge in my paper, however.

I would have to disagree that Craig and Plantinga are intellectually dishonest. I haven't seen as much of Plantinga as I have Craig, but I know Bill, and I see nothing roguish in the man's behavior or studies whatsoever. :)

First of all, I want to thank you for your consideration of my post and for your scholarship and input into this topic. My response to several points you raised follows, with your original text separated by bars.

//I'm afraid that the summary version of the ontological argument, as presented in the Wikipedia (your evident source for it), is neither historically nor conceptually very precise.//

Thanks for raising the objection. I think I understand what you mean in the discussion that follows: http://www.princeton.edu/~grosen/puc/phi203/ontological.html

If the presentation in that paper captures Anselm's original meaning better than the Wikipedia presentation, I will edit my post to include that presentation (as opposed to Wikipedia's). In the meantime, I have included in my documentation that the presentation is quite possibly not correct. I have also included your commentary in my citation of your work, so that your authored views are represented more clearly to readers here.

I should note that I think my example inherently demonstrates that imagining the existence of 'x' is not necessarily greater than or equal to the actual existence of 'x.' One can imagine an alien ship about to destroy the Earth, but if this ship necessarily exists, I think it would be a bad thing. Perhaps my understanding is more fundamentally flawed, however.

//I love the way you try to play the same dramatics with Plantinga's ontological argument. If you can find a hole in it, so much the worse for all other versions of the argument. That's pretty dicey isn't it?//

Actually, you're quite right. Just because my opinion that Plantinga's presentation is the most challenging Ontological Argument possible doesn't mean that it defeats those I find less challenging, or that it defeats some other forms in the literature which my comments here do not address.

In accordance, I've changed the post to strictly state that this defeats Plantinga's version of the argument, as per my opinion.

//And doesn't it get even scarier if that's the approach you use to make the case for the non-demonstrability of God's existence simpliciter? I'm less sure than you are that you have demolished a particular argument by finding a hole in one version you have selected, let alone that doing so takes you on the way to demonstrating the non-existence of God.//

Please be very careful here! My post specifically states that I cannot reach the existence of God via the Ontological Argument. Perhaps in my studies I would find that the Kalam Cosmological Argument works, that the book by N.T. Wright about Christ's Resurrection is solid, that Shabir Ally's Islamic theology is convincing, or the Dalai Lama's works on spirituality are sound. Like Anthony Flew, I would change my mind with little care for the consequences. Until then, I hold to nontheism, since I have not seen any proof that works.

I don't think there's anything in my post cheering on that I've made the case against God easier in any way, as you claim. The Ontological Argument is not a widely accepted argument, and, as I mentioned, even Bill Craig marginalizes it in the last few pages of the last chapter on his proofs for God.

Even given the assumption that I've set out to prove the impossibility of establishing God's existence, I've done no more than kick an extra point, since something like the Teleological or Moral Arguments are more widely used and accepted. But I am not here to do that. I am here as a skeptic analyzing the arguments for God as critically as I can, correcting and changing what mistakes I make when people comment on my posts, and then reaching the most sound conclusion possible.

//This has absolutely nothing to do with assigning numbers to greatness distributed over the available number of logically possible worlds. There is an issue here, but it has to do with the different versions of modal systems one can use in constructing or evaluating an argument.//

On the contrary: I am quite familiar with the objection, but I left it alone because I am not qualified to evaluate the validity of his chosen version of modal logic. It's been at least a decade since I've seen formal philosophical logic back in my undergrad days.

However, I submit that the incoherency of the methodology used to establish the inductive premise, as given by Plantinga and reviewed by Mr. Mears paper (listed in my citiations at the end), renders this premise faulty apart from the need to analyze the logical axiomatic assumptions underlying Plantinga's case for this premise. Although challenging Plantinga's modal axioms may be an easier route, I would invite you (not being snarky here!) to present a criticism that the mathematical approach that I and Mears took is by necessity irrelevant to the premise.

//What I'm getting at is this: As you are working through your project of "debunking Christianity," please make sure that you're not doing so at the cost of creditable scholarship.//

What's so odd is that I'm likely the single blogger here who is not writing to debunk Christianity, but to give as much of a definitive presentation of my critiques of theistic arguments as I can here on this blog. Many DC posters and perhaps a few constant readers know this.

The only versions of Christianity I'd ever set out to debunk are those which hold to theonomy or dominionism, something that even other Christians (such as Spurgeon) have attempted. I may post about those particular Christian worldviews in the future, but the majority of my posts here are directed toward the existence of God, which is one of the most important questions one can ponder. Although one may find it easier to simply believe, I weigh the metaphysical more than the epistemological, and the more important the question is in a metaphysical sense, the more critical my approach will be.

In the given context of my current knowledge, I maintain that I am still unconvinced.

// I have always said that if Christianity is true it should be able to withstand the hardest legitimate questions. Shouldn't the same criterion apply to a non-Christian world view? //

Of course it should! John and I didn't spend hundreds of dollars, nearly a week of our lives, and 3000+ miles driving to the Providence Apologetics Conference and meeting with people like Craig, Habermas, and Copan for the purpose of bragging or laughing at Christianity, you know. ;)

My scholarship in some particular subjects requires more reading and thought, admittedly. But those subjects which I am more qualified to present are posted here, with as much honesty to scholarship and dedication to discovering the truth as my mind allows. My friend John has graciously provided me with an opportunity to post my more thorough writings here despite any difference in ultimate purpose. The greatest result, to me, is an opportunity to view criticisms from both Christian and non-Christian worldviews alike in these commentaries, since this blog is out there to be read by scholars such as yourself.

I don't approach the believer's half of my library or all the archived debates here on the Internet with the predisposition of "I'm gonna destroy this!" I used to have that notion, and it led me to stop reading any sort of scholarly work, and if it weren't for honest people on both sides of the question knocking my skull about it, I'd be on one of Dawkins' buses thinking Robert Ingersoll did it all, not here begging for critiques of my work.

Yes, sometimes I err in my presentation of the scholarship, but I correct myself when I do. I've misrepresented nontheistic scholarship here before, too. But that's typically from a misstep in understanding the arguments or from a bit of sloppiness en route to a main point, not from any deliberate noncritical quote-mining used for the effect of dressing up some uncritical predisposition of mine with an appearance of scholarly citation. The important question of God's existence doesn't deserve that.

I look forward to the criticism more than the compliments - finding the facts is why I'm here. ;)

I would have to disagree that Craig and Plantinga are intellectually dishonest. I haven't seen as much of Plantinga as I have Craig, but I know Bill, and I see nothing roguish in the man's behavior or studies whatsoever. :)

They are both intelligent, learned and well-read but their (apologetic) papers contain fundamental logical/procedural flaws. Since they've managed to publish and achieve some respect within the community, perhaps these are indicative of a deplorable lack of standards within Apologetics or theology and Craig and Plantinga are merely prominent examples.

Within the scientific community, I think "intellectually dishonest" would be a mild criticism. So what would you call someone who publishes papers which lacks rigour, fails to define his terms, assumes what he tries to prove and fails to validate his assumptions? (Besides "a theologian" I mean :) )

BTW: I do like the limit examples. When I see maximum, I immediately think about limits and infinite series. The way it's phrased, "benevolence" appears to be a single-value or attribute but I suspect it's actually a function of many values: compassion, empathy, charity, equality, altruism, self-sacrifice, etc. I also suspect that there are bound to be trade-offs between the different traits. For instance, to be charitable requires taking food/money/resources from one person and giving it to another. If you give everything you own then you will die and so be less effective than someone who lives; if you rely on donations or taxes then to your donors you are a drain and not a help. I would also bet that there is a staggering amount of species-ism under the surface which is going unexplored. The focus is always on caring for humans but beetles are more numerous and have more species diversity yet no one ever thinks that to be maximally benevolent a person must show as much compassion towards beetles as to humans. Might an omni-benevolent being kill humans in order to preserve the last members of an endangered species of tree frogs? Sounds horrible but we don't blink at the thought of an omni-benevolent being killing a malaria-carrying mosquito. Even if we found some self-serving rationale why an omni-benevolent being would care only for humans, there is such diversity of opinion that there will be humans who will feel upset, oppressed or neglected for the same reasons we see when we consider different species.

It may be that there is one (or many) ways of maximizing your benevolence through different combinations of these traits but it will definitely be a trade-off and there will be people who will think this person/entity isn't such hot stuff. I would bet that if they had the, well, let's say "courage" to properly explore these questions they would see that "maximally great" wouldn't be omni-anything but rather very-. Very smart, very powerful, very benevolent. We say "omnipotent" but what exactly would this be? If it means "do whatever I imagine, poof, magic" then that's clearly impossible. It's left the continuum of "powerful" and is no longer a function which we can maximize. It would have to be pooffed into existence by magic. Perhaps maximally powerful actually means able to do work with the theoretical minimum of energy loss. A 5000 mpg car may be maximally powerful, say. If this is the case, there is no reason to think that a maximally powerful or efficient car must exist in all possible worlds. It isn't omni-powerful, just maximally powerful.

When you break it down into the details, "maximally great" loses the halo and blind adoration which P/C use to turn "him" into "Him" and argue He exists in every possible world.

Barefoot Bum has it right. I also think, though, that it's a bit unfair to cast maximal greatness as numerical: being maximally great in a world Wx is explicitly not a matter of having the largest greatness value in that world.

Consider, for instance, a godless world that nonetheless has one entity - call it a person just for the sake of naming it - that's more powerful, knowing, and moral than any other entity in that world. For Plantinga (and Craig, if he's using the same kind of argument), that person would just be the greatest in that world, not maximally great. While maximal greatness implies being greater than some number of other things (whenever other things exist, that is), being greater than other things is not in any sense a necessary condition of maximal greatness in this sense: as you yourself say, in the (hypothetical) world with nothing but God in it, God is still maximally great despite having a greatness factor of 0. Whatever numerical oddities you can produce, then, just reinforce the idea that greatness can't sensibly be talked about numerically.

Does a maximally great being exist in every logically possible world,including worlds that contain vast amounts of pointless suffering?

I think that's in part two of Plantinga's argument. He's demonstrated that an omni-max God must exist everywhere if it's possible for a God to exist anywhere, so now we can conclude that since an omni-max God doesn't exist in our universe it must be impossible for such a God to exist anywhere.

Look for it in his upcoming series "More of God's Greatest Mistakes", "Just Who Is this God Person Anyway?" and "Well, That Just About Wraps it Up for God."

Steven - sure. since a world can be possible, in a hypothetical way that has no relationship to reality, you can make it be whatever way you want. Toss in some mathematical-style logical arguments, and you can obscure the point that none of the argument maps onto reality. As soon as Plantiga can show that any one of these possible worlds has an actual (as opposed to a conceptual) existence, then we might take his arguments more seriously than mental masturbation. And that doesn't even take care of his other flaws that some have mentioned.

Seriously, just put the word "I" in place of the god part of the argument, and you can show that you, in fact, are God. Since it is possible that I exist in all possible, worlds, and that in one of them I am maximally greatest (which therefore means that I am that way in all possibilities somehow), this means that I am God, but for some reason I don't have my power and knowledge now. Perhaps I turned it off in an effort to ensure that mankind has free will.

The wording of that first premiss has always irked me. Why isn't it written such that you don't have to turn your brain inside out to get it: God is that entity greater than which nothing can be conceived?

I had a couple of thoughts on this. Firstly, is his argument that it is possible for God to exist, it is not possible that he doesn't exist?Secondly, can it be re-phrased to argue for the opposite conclusion?1. It is possible that no maximally great being exists.2. If it is possible that no maximally great being exists, then there must be some possible world in which no maximally great being exists.3. If there is a possible world with no maximally great being, then he cannot exist in any possible world4. If he does not exist in any possible world, he cannot exist in the real world.

To be honest, I just don't understand why Plantinga's Modal Ontological Argument is taken seriously by professional philosophers. The metaphysical quibbling over possible worlds, omni-whatnot and excellence is beside the point: the argument is simply logically invalid.

An ordinary person can be forgiven for not spotting the invalidity, but professional philosophers are supposed to be experts at formal logical reasoning. Critiquing Plantinga's argument on metaphysical grounds is like critiquing the choice of materials for plumbing a house without noticing that the plumber has failed to install U-traps on the drains.

"any infinite sum, when the elements of the sum are reordered to the effect that the first term is greatest, the second term is at most the value of the first, the third at most the value of the second, etc., then the limit of the terms themselves must be zero as we tend toward infinity"

Um, for some constant a let S(n)=a+10^(-n)

for example, let a be 5. you have an infinite, monotonic decreasing sequence as follows: 5.1, 5.01, 5.001, 5.0001, ... which certainly does not converge to zero.